Consuming Love The fanlisting for the relationship between Damon & Elena
You want a love that consumes you.
    You want passion and adventure, and even a little danger...
I want you to get everything you're looking for.
— Damon to Elena

Elena & Damon

Damon first met Elena the night her parents died. She met Damon in the middle of the road; at first, he mistook her for Katherine. They started talking and even flirted a little. He told her that she wanted a love that consumed her, a love that was passionate, and a love that was a little dangerous. And then he told her that he wanted her to have everything she wanted. After their moonlit conversation, Elena's parents had arrived and Damon compelled her to forget it ever happened. They were later reunited in The Night of the Comet, but she didn't remember him. This also explains the reason how Damon knew who Elena was when they met in the Salvatore Boarding House. Elena admitted in The Departed, that if she had met Damon first, things might have gone different between them and she would have chosen him over Stefan. This memory resurfaces after Elena transitioned into a vampire in Growing Pains.

Canon

In the first season of the show, Damon and Elena develop an interesting bond. Though Elena only sees him as a friend or even an enemy at times, he still decides to give into his feelings for her, especially after he discovers that his former lover Katherine is not sealed in a tomb as he had expected her to be. But for the time being Elena is still hung up on Damon's brother Stefan.

Damon and Elena's relationship starts off bad, in the beginning of the second season. After Damon was deceived by Katherine, into believing he had actually kissed Elena, he was very hurt when Elena stated that she would never have kissed him back. Later, Damon went to Elena's house and tried to convince her that she was wrong about her feelings for him. When she insists upon her words, Damon lashes out and kills Elena's brother Jeremy (who came back to life with the Gilbert ring). After that, Elena does not forgive him easily, but eventually she does forgive him, only to get mad at him again, for giving her his blood, to save her from Klaus' sacrifice. But all is forgiven in the last episode of season two, when Damon almost dies from wolf venom and Elena is there for him and she even kisses him. All is ruined though when they finally find out that Stefan had given himself over to Klaus.

In the third season, Damon and Elena are like best friends. They helped each other heal over the summer, in which Stefan had disappeared with Klaus. Elena kept bringing him new leads as to where Stefan might be but he chooses to investigate them on his own and keep her in the dark, because she doesn't want her to know that Stefan has been the one killing people. On the night of her birthday, he gives her the necklace she had lost,which had originally been o gift from Stefan. As it is discovered later in the show,that was the moment she knew that she loved him. They then continue chasing after Stefan and Damon is always by Elena's side when she needs comfort and vice versa. After Klaus makes Stefan turn his humanity off and feed on Elena, Damon vowed to never leave her again. A little later from when Elena had Damon take away Jeremy's memories, they finally share their first kiss outside her house. Things get rough when Stefan gets his humanity back and Elena has to choose between the two brothers. When Elena and Damon are stuck in a motel in Denver, they make out again. This time being the first Elena actually initiated it. But inspire of all, Elena picks Stefan in the end and tells Damon that thing would be different if they had met first. When Elena falls off the Wickery bridge she starts her transition to a vampire and gets her memory of their first meeting back.

In the fourth season Elena becomes a vampire, something that changed the was she saw life and her perspective completely. So at first she tries to deal with her intense feelings, but gives up at some point and breaks up with Stefan over Damon. Shortly after that the two sleep together, while at the same time Stefan and Caroline figure out that Elena is actually sired to Damon. When Elena believes her feelings for him are real, he along with everyone else are doubtful. She even tells him that she loves him, enough to let him take the cure to vampirism with her which he declines. Then due to her brother's death, Elena turns her humanity off and Damon tries everything to bring her back and he along with Stefan eventually succeed. Then it is confirmed by her that her feelings were true since the sire bond no longer existed and they start dating.

During the next seasons they go through hell but they get through it and eventually get together but their biggest hardship came when Kai linked Elena's life to Bonnie's and as long as she were alive, Elena would remain asleep. But the two are reunited again and in a flash forward scene its is shown that they were married in the series finale.

-info from The Vampire Diaries Wiki & Shipping Wiki